On Meromorphic Parabolic Starlike Functions with Fixed Point Involving the q-Hypergeometric Function and Fixed Second Coefficients

Abstract

This article defines a new class of meromorphic parabolic starlike functions in the punctured unit disc $D^{*}=\{z\in \mathbb{C}:0<|z|<1\}$ that includes fixed second coefficients of class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right)$ and the q- hypergeometric functions. For the function belonging to the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right),$ some properties are obtained, including the coefficient inequalities, closure theorems, and the radius of convexity.

1. Introduction

Let $\eta$ be a fixed point in the unit disc $D:=\{z\in \mathbb{C}:|z|<1\}$. Using $\mathcal{H}\left(D\right)$, denote the class of functions that are regular and

$$\mathcal{A}\left(\eta \right)=\{f\in \mathcal{H}\left(D\right):f\left(\eta \right)=f^{{}^{\prime }}\left(\eta \right)-1=0\}$$

Using ${\mathcal{S}}_{\eta }=\{f\in \mathcal{A}\left(\eta \right),denotethefollowing:f$ is univalent in D}, the subclass of $A\left(\eta \right)$ consisting of the functions of the form

$$\zeta \left(z\right)=z-\eta +\sum _{l=1}^{\infty }{a}_l{(z-\eta )}^l.$$

Let $\varkappa$ denote the class of meromorphic functions $\zeta \left(z\right)$ of the form

$$\zeta \left(z\right)=\frac{1}{z}+\sum _{l=1}^{\infty }{a}_l{z}^l$$

(1)

defined on the punctured unit disc $D^{*}=\{z\in \mathbb{C}:0<|z|<1\}$.

Using ${\varkappa }_{\eta }$, denote the subclass of $\varkappa$ consisting of the form’s functions

$$\zeta \left(z\right)=\frac{1}{z-\eta }+\sum _{l=1}^{\infty }{a}_l{(z-\eta )}^l,a_l\ge 0;z\ne \eta ,z\in D.$$

(2)

If a function $\zeta \left(z\right)$ of the form (2) belongs to the class of meromorphic starlike of order $\sigma$$(0\ge \sigma <1)$, it is indicated by ${\varkappa }_{\eta }^{*}\left(\sigma \right)$, if

$$-\Re \left(\frac{\left(z-\eta \right){\zeta }^{{}^{\prime }}\left(z\right)}{\zeta \left(z\right)}\right)>\sigma ,z\ne \eta ,z\in D,$$

(3)

and belongs to a class of meromorphic convex of order $\sigma (0\le \sigma <1)$, which is indicated by ${\varkappa }_{\eta }^k\left(\sigma \right)$, if

$$-\Re \left(1+\frac{\left(z-\eta \right){\zeta }^{{}^{\prime \prime }}\left(z\right)}{{\zeta }^{{}^{\prime }}\left(z\right)}\right)>\sigma ,z\ne \eta ,z\in D.$$

(4)

For functions $\zeta \left(z\right)$, given by (2) and $g\left(z\right)=\frac{1}{z-\eta }+{\sum }_{l=1}^{\infty }{b}_l{\left(z-\eta \right)}^l$, $(b_l\ge 0)$, we define the Hadamard product or convolution of $\zeta \left(z\right)$ and $g\left(z\right)$ by

$$\left(\zeta \ast g\right)\left(z\right)=\frac{1}{z-\eta }+\sum _{l=1}^{\infty }{a}_l{b}_l{\left(z-\eta \right)}^l=(g\ast \zeta )\left(z\right).$$

(5)

Define the following operator [1].

$$q_{\mu ,\xi }\left(z\right)=\frac{1}{z-\eta }+\sum _{l=1}^{\infty }{\left(\frac{\mu }{l+1+\mu }\right)}^{\xi }{\left(z-\eta \right)}^l,(\mu >0,\xi \ge 0).$$

(6)

Cho [2], Ghanim, and Darus [3] studied the above function when $\eta =0$.

Corresponding to the function $q_{\mu ,\xi }\left(z\right)$ and using the Hadamard product for $\zeta \left(z\right)\in {\varkappa }_{\eta },$ we define a new linear operator $\mathcal{J}\left(\mu ,\xi ,\eta \right)$ on ${\varkappa }_{\eta }\left(\sigma \right)$ by

$${\mathcal{J}}_{\mu ,\xi }\zeta \left(z\right)=(\zeta \left(z\right)\ast q_{\mu ,\xi }\left(z\right))=\frac{1}{z-\eta }+\sum _{l=1}^{\infty }{\left(\frac{\mu }{l+1+\mu }\right)}^{\xi }\left|a_l\right|{\left(z-\eta \right)}^l.$$

(7)

When $\eta =0$, it reduces to Ghanim and Darus [4].

A generalized q-Taylars formula for fractional q-calculus was introduced more recently by Purohit and Raina [5], who also derived a few q-generating functions for q-hypergeometric functions.

As with the aforementioned functions, we attempt to derive a generalized differential operator on meromorphic functions in $D^{*}=\{z\in \mathbb{C}:0<|z|<1\}$ in this paper and study some of their characteristics.

For complex parameters ${\gamma }_1,\dots ,{\gamma }_d$ and ${\beta }_1,\dots ,{\beta }_s({\beta }_t\ne 0,-1,\dots ;t=1,2,\dots ,s)$ the q-hypergeometric function ${}_d{\mathsf{\Phi }}_s\left(z\right)$ is defined by

$${}_d{\mathsf{\Phi }}_s({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s;q,z)=\sum _{l=0}^{\infty }{\displaystyle \frac{{({\gamma }_1,q)}_l\dots {({\gamma }_d,q)}_l}{{(q,q)}_l{({\beta }_1,q)}_l\dots {({\beta }_s,q)}_l}}\times {[{(-1)}^l{q}^{\left(\begin{array}{c}l\\ 2\end{array}\right)}]}^{1+s-d}{z}^l,$$

(8)

with $\left(\begin{array}{c}l\\ 2\end{array}\right)=l(l-1)/2$ where $q\ne 0$ when $d>s+1(d,s\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\};z\in D^{*}).$ The q-shifted factorial is defined for $\gamma ,q\in \mathbb{C}$ as a product of l factors by

$${\left(\gamma ;q\right)}_l=\left\{\begin{array}{cc}\left(1-\gamma \right)\left(1-\gamma q\right)\dots \left(1-\gamma q^{l-1}\right)\hfill & \left(l\in \mathbb{N}\right)\hfill \\ 1\hfill & \left(l=0\right)\hfill \end{array}\right.$$

(9)

and in terms of basic analogue of the gamma function

$${\left(q^{\gamma },q\right)}_l={\displaystyle \frac{{\mathsf{\Gamma }}_q\left(\gamma +l\right){\left(1-q\right)}^l}{{\mathsf{\Gamma }}_q\left(\gamma \right)}}l>0.$$

(10)

It is important to note that ${lim}_{q\to -1}\left({(q^{\gamma };q)}_l/{\left(1-q\right)}^l\right)={\left(\gamma \right)}_l=\gamma (\gamma +1)\cdots (\gamma +l-1)$ is the familiar Pochhammer symbol and

$${}_d{\mathsf{\Phi }}_s\left({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s;z\right)=\sum _{l=0}^{\infty }{\displaystyle \frac{{\left({\gamma }_1\right)}_l\dots {\left({\gamma }_d\right)}_l}{{\left({\beta }_1\right)}_l\dots {\left({\beta }_s\right)}_l}}{\displaystyle \frac{z^l}{l!}}.$$

(11)

Now, for $z\in D,0<\left|q\right|<1,$ and $d=s+1$, the basic hypergeometric function defined in (8) takes the form

$${}_d{\mathsf{\Phi }}_s\left({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s,;q,z\right)=\sum _{l=0}^{\infty }{\displaystyle \frac{{\left({\gamma }_1,q\right)}_l\dots {\left({\gamma }_d,q\right)}_l}{{\left(q,q\right)}_l{\left({\beta }_1,q\right)}_l\dots {({\beta }_s,q)}_l}}{z}^l,$$

(12)

which converges absolutely in the open disc D.

According to the recently introduced function ${}_d{\mathsf{\Phi }}_s({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s;q,z)$ for meromorphic functions $\zeta \in \varkappa$ consisting of functions of the form (1), Al-dweby and Darus [6] developed the q-analogue of the Liu-Srivastava operator, as follows:

$$\begin{array}{c}{}_d{\mathsf{{\rm Y}}}_s\left({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s,;q,z\right)\ast \zeta \left(z\right)={\displaystyle \frac{1}{z}}{}_d{\mathsf{\Phi }}_s({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s,;q,z)\ast \zeta \left(z\right)\\ ={\displaystyle \frac{1}{z}}+\sum _{l=1}^{\infty }{\displaystyle \frac{{\prod }_{i=1}^d{\left({\gamma }_i,q\right)}_{l+1}}{{\left(q,q\right)}_{l+1}{\prod }_{i=1}^s{\left({\beta }_i,q\right)}_{l+1}}}{a}_l{z}^l,\end{array}$$

(13)

where ${\prod }_{n=1}^m{({\gamma }_n,q)}_{l+1}={({\gamma }_1,q)}_{l+1}{({\gamma }_2,q)}_{l+1}\cdots {({\gamma }_m,q)}_{l+1}$, where $z\in D^{*}=\{z\in \mathbb{C}:0<|z|<1\},$ and

$$\begin{array}{c}{}_d{\mathsf{{\rm Y}}}_s({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s,;q,z)={\displaystyle \frac{1}{z}}{}_d{\mathsf{\Phi }}_s({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s,;q,z)\\ ={\displaystyle \frac{1}{z}}+\sum _{l=1}^{\infty }{\displaystyle \frac{{\prod }_{i=1}^d{\left({\gamma }_i,q\right)}_{l+1}}{{\left(q,q\right)}_{l+1}{\prod }_{i=1}^s{\left({\beta }_i,q\right)}_{l+1}}}{z}^l.\end{array}$$

(14)

Murugusundaramoorthy and Janani [7] defined the following linear operator for functions $\zeta \in {\varkappa }_{\eta }$ and for real parameters ${\gamma }_1,\dots ,{\gamma }_d$ and ${\beta }_1,\dots ,{\beta }_s({\beta }_t\ne 0,-1,\dots ;t=1,2,\dots ,s)$:

$${}_d{\mathsf{{\rm Y}}}_s\left({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s,;q,z-\eta \right):{\varkappa }_{\eta }\to {\varkappa }_{\eta },$$

(15)

$$\begin{array}{c}{}_d{\mathsf{{\rm Y}}}_s\left({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s,;q,z-\eta \right)={\displaystyle \frac{1}{z-\eta }}{}_d{\mathsf{\Phi }}_s({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s,;q,z-\eta )\\ ={\displaystyle \frac{1}{z-\eta }}+\sum _{l=1}^{\infty }{\displaystyle \frac{{\prod }_{i=1}^d{\left({\gamma }_i,q\right)}_{l+1}}{{\left(q,q\right)}_{l+1}{\prod }_{i=1}^s{\left({\beta }_i,q\right)}_{l+1}}}{(z-\eta )}^l.\end{array}$$

(16)

Corresponding to the functions ${}_d{\mathsf{{\rm Y}}}_s\left({\gamma }_1,\dots ,{\gamma }_d;{\beta }_1,\dots ,{\beta }_s,;q,z-\eta \right)$ and $q_{\mu ,\xi }\left(z\right)$ given in (6) and using the Hadamard product for $\zeta \left(z\right)\in {\varkappa }_{\eta },$ we define a new linear operator ${\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_1,{\gamma }_2,\dots {\gamma }_d;{\beta }_1,{\beta }_2,\dots {\beta }_s;q\right)$ on ${\varkappa }_{\eta }$ by

$${\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_1,{\gamma }_2,\dots {\gamma }_d;{\beta }_1,{\beta }_2,\dots {\beta }_s;q\right)\zeta \left(z\right)$$

$$=\left(\zeta \left(z\right){\ast }_d{\mathsf{{\rm Y}}}_s({\gamma }_1,{\gamma }_2,\dots {\gamma }_d;{\beta }_1,{\beta }_2,\dots {\beta }_s;q,z-\eta )\ast q_{\lambda ,\xi }\left(z\right)\right)$$

(17)

$$\begin{array}{c}\hfill ={\displaystyle \frac{1}{z-\eta }}+\sum _{l=1}^{\infty }{\displaystyle \frac{{\prod }_{i=1}^d{\left({\gamma }_i,q\right)}_{l+1}}{{\left(q,q\right)}_{l+1}{\prod }_{i=1}^s{\left({\beta }_i,q\right)}_{l+1}}}{\left({\displaystyle \frac{\mu }{l+1+\mu }}\right)}^{\xi }{a}_l{(z-\eta )}^l.\end{array}$$

$$={\displaystyle \frac{1}{z-\eta }}+\sum _{l=1}^{\infty }{\mathsf{\Omega }}_s^d\left(l\right)a_l{(z-\eta )}^l,$$

(18)

where

$${\mathsf{\Omega }}_s^d\left(l\right)={\displaystyle \frac{{\prod }_{i=1}^d{\left({\gamma }_i,q\right)}_{l+1}}{{\left(q,q\right)}_{l+1}{\prod }_{i=1}^s{\left({\beta }_i,q\right)}_{l+1}}}{\left({\displaystyle \frac{\mu }{l+1+\mu }}\right)}^{\xi }.$$

(19)

For convenience, we will denote

$${\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_1,{\gamma }_2,\dots {\gamma }_d;{\beta }_1,{\beta }_2,\dots {\beta }_s;q\right)\zeta \left(z\right)={\mathcal{J}}_{\xi }^{\mu }({\gamma }_d,{\beta }_s,q)\zeta \left(z\right).$$

(20)

In (17), for $\xi =0$, the operator was investigated by Murugusundaramoorthy and Janani [7].

Recent studies on the meromorphic functions with generalized hypergeometric functions and with q-hypergeometric functions include those by Cho and Kim [8], Dziok and Srivastava [9,10], Ghanim [11], Ghanim et al. [12,13], Liu and Srivastava [14,15], Aldweby and Darus [6], Murugusundaramoorthy and Janani [7]. We define the following new subclass of functions in ${\varkappa }_{\eta }$ using the generalized operator ${\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right).$ In response to earlier work on meromorphic functions by function theorists (see [15,16,17,18,19,20,21,22]).

For $0\le \nu <1$ and $0\le \psi \le 1$, we let ${\mathcal{A}}_s^d(\psi ,\tau ,\nu ,\eta )$ indicate a subclass of ${\varkappa }_{\eta }$ that consists of functions of the form (2) that satisfy the requirement that

$$-\Re \left({\displaystyle \frac{\left(z-\eta \right){\left({\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)\right)}^{{}^{\prime }}+\psi {(z-\eta )}^2{\left({\mathcal{J}}_{\xi }^{\mu }({\gamma }_d,{\beta }_s,q)\zeta \left(z\right)\right)}^{\prime \prime }}{\left(1-\psi \right){\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)+\psi (z-\eta ){\left({\mathcal{J}}_{\xi }^{\mu }({\gamma }_d,{\beta }_s,q)\zeta \left(z\right)\right)}^{\prime }}}\right)$$

$$>\tau \left|{\displaystyle \frac{\left(z-\eta \right){\left({\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)\right)}^{{}^{\prime }}+\psi {(z-\eta )}^2{\left({\mathcal{J}}_{\xi }^{\mu }({\gamma }_d,{\beta }_s,q)\zeta \left(z\right)\right)}^{\prime \prime }}{\left(1-\psi \right){\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)+\psi (z-\eta ){\left({\mathcal{J}}_{\xi }^{\mu }({\gamma }_d,{\beta }_s,q)\zeta \left(z\right)\right)}^{\prime }}}+1\right|+\nu ,$$

(21)

where (17) is used to give ${\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)$.

Additionally, we can state this condition by

$$-\Re \left(\frac{\left(z-\eta \right)F^{{}^{\prime }}\left(z\right)}{F\left(z\right)}\right)>\tau \left|{\displaystyle \frac{\left(z-\eta \right)F^{{}^{\prime }}\left(z\right)}{F\left(z\right)}}+1\right|+\nu ,$$

(22)

where

$$F\left(z\right)=\left(1-\psi \right){\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)+\psi \left(z-\eta \right){\left({\mathcal{J}}_{\xi }^{\mu }({\gamma }_d,{\beta }_s,q)\zeta \left(z\right)\right)}^{\prime }$$

$$={\displaystyle \frac{1-2\psi }{z-\eta }}+\sum _{l=1}^{\infty }\left(l\psi -\psi +1\right){\mathsf{\Omega }}_s^d\left(l\right)a_l{\left(z-\eta \right)}^l,a_l\ge 0$$

(23)

where ${\mathsf{\Omega }}_s^d\left(l\right)$ defind by (18).

It is interesting to note that we can define a number of new subclasses of ${\varkappa }_{\eta }$ by specializing the parameters $\psi ,\tau$ and $d,s$. In the examples that follow, we demonstrate two significant subclasses.

Example 1.

For $\psi =0$, we let ${\mathcal{A}}_s^d(0,\tau ,\nu ,\eta )={\mathcal{A}}_s^d\left(\tau ,\nu ,\eta \right)$ indicate a subclass of ${\varkappa }_{\eta }$ that consists of functions of the form (2), which satisfy the requirement that

$$-\Re \left({\displaystyle \frac{\left(z-\eta \right){\left({\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)\right)}^{{}^{\prime }}}{{\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)}}\right)>\tau \left|{\displaystyle \frac{\left(z-\eta \right){\left({\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)\right)}^{{}^{\prime }}}{{\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)}}+1\right|+\nu ,$$

(24)

where ${\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)$ is given by (17).

Example 2.

For $\psi =0,\tau =0,$ we let ${\mathcal{A}}_s^d\left(0,0,\nu ,\eta \right)={\mathcal{A}}_s^d\left(v,\eta \right)$ indicate a subclass of ${\varkappa }_{\eta }$ that consists of functions of form (2) that satisfy the requirement that

$$-\Re \left({\displaystyle \frac{\left(z-\eta \right){\left({\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)\right)}^{{}^{\prime }}}{{\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)}}\right)>\nu ,$$

(25)

where ${\mathcal{J}}_{\xi }^{\mu }\left({\gamma }_d,{\beta }_s,q\right)\zeta \left(z\right)$ is given by (17).

We begin by recalling the following lemma due to Challab, Darus and Ghanim [1].

Lemma 1

([1]). The function $\zeta \left(z\right)$ defined by (2) is in the class in ${\mathcal{A}}_s^d(\psi ,\tau ,\nu ,\eta )$ if, and only if,

$$\sum _{l=1}^{\infty }\left[l\left(1+\tau \right)+\left(\nu +\tau \right)\left]\right(1+l\psi -\psi \right){\mathsf{\Omega }}_s^d\left(l\right)a_l\le (1-2\psi )(1-\nu ).$$

(26)

The result is sharp.

In view of Lemma 1, we can see that the function $\zeta \left(z\right)$, defined by (2) in the class ${\mathcal{A}}_s^d(\psi ,\tau ,\nu ,\eta )$, satisfies the ceofficient inequality

$$a_1\le {\displaystyle \frac{\left(1-2\psi \right)\left(1-\nu \right)}{\left[\left(1+v+2\tau \right)\right]{\mathsf{\Omega }}_s^d\left(1\right)}},$$

(27)

where

$${\mathsf{\Omega }}_s^d\left(1\right)={\displaystyle \frac{{\prod }_{i=1}^d{\left({\gamma }_i,q\right)}_2}{{\left(q,q\right)}_2{\prod }_{i=1}^s{\left({\beta }_i,q\right)}_2}}{\left({\displaystyle \frac{\mu }{\mu +2}}\right)}^{\xi }.$$

Hence we may take

$$a_1={\displaystyle \frac{\left(1-2\psi \right)\left(1-\nu \right)c}{\left[\left(\left(1+v+2\tau \right)\right)\right]{\mathsf{\Omega }}_s^d\left(1\right)}}0\le c\le 1.$$

(28)

Making use of (27), we now introduce the following class of functions:

Let ${\mathcal{A}}_{s,c}^d(\psi ,\tau ,\nu ,\eta )$ denote the subclass of ${\mathcal{A}}_s^d(\psi ,\tau ,\nu ,\eta )$, consisting of a function of the form

$$\zeta \left(z\right)={\displaystyle \frac{1}{z-\eta }}+{\displaystyle \frac{(1-2\psi )(1-\nu )c}{\left[(1+\nu +2\tau )\right]{\mathsf{\Omega }}_s^d\left(1\right)}}(z-\eta )+\sum _{l=2}^{\infty }{a}_l{\left(z-\eta \right)}^l,$$

(29)

where

$$a_{l\ge 0}and0\le c\le 1.$$

In this paper, we obtain the coefficient inequalities for the class ${\mathcal{A}}_{s,c}^d(\psi ,\tau ,\nu ,\eta )$ and closure theorems. Further, the radius of convexity are obtained for the class ${\mathcal{A}}_{s,c}^d(\psi ,\tau ,\nu ,\eta )$.

2. Coefficients’ Inequalities

Theorem 1.

Let the function $\zeta \left(z\right)$ be defined (28). Then, $\zeta \left(z\right)$ is in the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right)$ if, and only if,

$$\sum _{l=2}^{\infty }\left[l\left(1+\tau \right)+\left(\nu +\tau \right)\left]\right(1+l\psi -\psi \right){\mathsf{\Omega }}_s^d\left(l\right)a_l\le (1-2\psi )(1-\nu )(1-c).$$

(30)

The result is sharp.

Proof. 

Putting

$$a_1={\displaystyle \frac{\left(1-2\psi \right)\left(1-\nu \right)c}{[\left(1+\tau \right)+\left(v+\tau \right)]{\mathsf{\Omega }}_s^d\left(1\right)}}0\le c\le 1,$$

(31)

Using (25) and simplification, we arrive at the result, which is sharp for the function

$$\zeta \left(z\right)=\frac{1}{z-\eta }+\frac{(1-2\psi )(1-\nu )c}{[(1+\tau )+(\nu +\tau )]{\mathsf{\Omega }}_s^d\left(1\right)}(z-\eta )+\frac{(1-2\psi )(1-\nu )(1-c)}{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)}{(z-\eta )}^l(l\ge 2).$$

(32)

    □

Corollary 1.

Let the function $\zeta \left(z\right)$ defined by (27) be in the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right).$ Then,

$$a_l\le {\displaystyle \frac{(1-2\psi )(1-\nu )(1-c)}{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)}}(l\ge 2).$$

(33)

The result for the function $\zeta \left(z\right)$ given by (31) is sharp.

Corollary 2.

If $0\le c_1\le c_2\le 1$

$${\mathcal{A}}_{s,c_2}^d\left(\psi ,\tau ,\nu ,\eta \right)\subseteq {\mathcal{A}}_{s,c_1}^d\left(\psi ,\tau ,\nu ,\eta \right).$$

3. Closure Theorems

Using Theorem 1, we can prove the following theorems:

Theorem 2.

Let the function

$${\zeta }_j\left(z\right)=\frac{1}{z-\eta }+\frac{(1-2\psi )(1-\nu )c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}(z-\eta )+\sum _{l=2}^{\infty }{a}_{l,j}{\left(z-\eta \right)}^l(a_{l,j}\ge 0),$$

(34)

be in the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right).$ for every $j=1,2,\dots ,s.$ Then, the function

$$g\left(z\right)={\displaystyle \frac{1}{z-\eta }}+{\displaystyle \frac{(1-2\psi )(1-\nu )c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}(z-\eta )+\sum _{l=2}^{\infty }{b}_l{\left(z-\eta \right)}^l(b_l\ge 0),$$

(35)

is also in the same class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right),$ where

$$b_l={\displaystyle \frac{1}{m}}\sum _{j=1}^m{a}_{l,j}(l=1,2,\dots ).$$

(36)

Proof. 

Since ${\zeta }_j\left(z\right)\in \mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right),$ it follows from Theorem 1 that

$$\sum _{l=2}^{\infty }\left[l\left(1+\tau \right)+\left(\nu +\tau \right)\left]\right(1+l\psi -\psi \right){\mathsf{\Omega }}_s^d\left(l\right)a_{l,j}\le (1-2\psi )(1-\nu )(1-c),$$

(37)

for every $j=1,2,\dots ,m.$ Hence,

$$\begin{array}{cc}\hfill & \sum _{l=2}^{\infty }\left[l\left(1+\tau \right)+\left(\nu +\tau \right)\left]\right(1+l\psi -\psi \right){\mathsf{\Omega }}_s^d\left(l\right)b_l\hfill \\ \hfill =& \sum _{l=2}^{\infty }\left[l\left(1+\tau \right)+\left(\nu +\tau \right)\left]\right(1+l\psi -\psi \right){\mathsf{\Omega }}_s^d\left(l\right)\left({\displaystyle \frac{1}{m}}\sum _{j=1}^m{a}_{l,j}\right)\hfill \end{array}$$

$$={\displaystyle \frac{1}{m}}\sum _{j=1}^m\left(\sum _{l=2}^{\infty }\left[l\left(1+\tau \right)+\left(\nu +\tau \right)\left]\right(1+l\psi -\psi \right){\mathsf{\Omega }}_s^d\left(l\right)a_{l,j}\right)$$

(38)

$$\begin{array}{cc}\hfill & \le (1-2\psi )(1-\nu )(1-c).\hfill \end{array}$$

From Theorem 1, it follows that $g\left(z\right)\in \mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right).$ This completes the proof.    □

Theorem 3.

The class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right)$ is closed under convex linear combination

Proof. 

Let ${\zeta }_j\left(z\right)(j=1,2)$ be defined by (33)

$$h\left(z\right)=\varphi {\zeta }_1\left(z\right)+(1-\varphi ){\zeta }_2\left(z\right)(0\le \varphi \le 1),$$

(39)

It is sufficient to prove that the function h(z) is also in the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right).$

Since

$$h\left(z\right)=\frac{1}{z-\eta }+\frac{(1-2\psi )(1-\nu )c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}(z-\eta )+\sum _{l=2}^{\infty }\left[\varphi a_{l,1}+(1-\varphi )a_{l,2}\right]{\left(z-\eta \right)}^l.$$

(40)

Then, we have Theorem 1, that

$$\begin{array}{cc}\hfill & \sum _{l=2}^{\infty }[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)\left[\varphi a_{l,1}+(1-\varphi )a_{l,2}\right]\hfill \\ \\ & \le \varphi (1-2\psi )(1-\nu )(1-c)+(1-\varphi )(1-2\psi )(1-\nu )(1-c)\hfill \\ \\ \hfill & =(1-2\psi )(1-\nu )(1-c).\hfill \end{array}$$

Therefore, $h\left(z\right)\in \mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right).$    □

Theorem 4.

Let

$$\zeta \left(z\right)={\displaystyle \frac{1}{z-\eta }}+{\displaystyle \frac{(1-2\psi )(1-\nu )c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}(z-\eta ),$$

(41)

and

$${\zeta }_l\left(z\right)={\displaystyle \frac{1}{z-\eta }}+{\displaystyle \frac{(1-2\psi )(1-\nu )c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}(z-\eta )+{\displaystyle \frac{(1-2\psi )(1-\nu )(1-c)}{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)}}{\left(z-\eta \right)}^l(l\ge 2).$$

(42)

Then, $\zeta \left(z\right)$ is in the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right),$ if, and only if, it can be expressed in the form

$$\zeta \left(z\right)=\sum _{l=1}^{\infty }{\lambda }_l{\zeta }_l\left(z\right),$$

(43)

where ${\lambda }_l\ge 0$ and ${\sum }_{l=1}^{\infty }{\lambda }_l=1.$

Proof. 

Let

$$\zeta \left(z\right)=\sum _{l=1}^{\infty }{\lambda }_l{\zeta }_l\left(z\right),$$

(44)

$$={\displaystyle \frac{1}{z-\eta }}+{\displaystyle \frac{(1-2\psi )(1-\nu )c}{[1+\nu +2\tau )]{\mathsf{\Omega }}_s^d\left(1\right)}}(z-\eta )+\sum _{l=2}^{\infty }{\displaystyle \frac{(1-2\psi )(1-\nu )(1-c)}{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)}}{\lambda }_l{\left(z-\eta \right)}^l.$$

Since

$$\sum _{l=2}^{\infty }{\displaystyle \frac{(1-2\psi )(1-\nu )(1-c){\lambda }_l}{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)}}.{\displaystyle \frac{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)}{(1-2\psi )(1-\nu )}}$$

(45)

$$=\left(1-c\right){\sum }_{l=2}^{\infty }{\lambda }_l=\left(1-c\right)\left(1-{\lambda }_1\right)\le \left(1-c\right).$$

Hence, using Theorem 1, we have $\zeta \left(z\right)\in {\mathcal{A}}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right).$

Conversely, we assume that $\zeta \left(z\right)$, defined by (28), is in the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right).$

Then by applying (32), we can obtain

$$a_l\le {\displaystyle \frac{(1-2\psi )(1-\nu )(1-c)}{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)}}(l\ge 2).$$

(46)

Setting

$${\lambda }_l={\displaystyle \frac{[l\left(1+\tau \right)+\left(\nu +\tau \right)]\left(1+l\psi -\psi \right){\mathsf{\Omega }}_s^d\left(l\right)}{\left(1-2\psi \right)\left(1-\nu \right)\left(1-c\right)}}{a}_l(l\ge 2).$$

(47)

$${\lambda }_1=1-\sum _{l=2}^{\infty }{\lambda }_l.$$

(48)

We have (42). The proof of Theorem 4 is now complete.    □

4. Radius of Convexity

Theorem 5.

Let the function $\zeta \left(z\right)$ be defined by (28) in the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right).$ Then, $\zeta \left(z\right)$ is meromorphically convex of order $\delta (0\le \delta <1)$ in $0<\left|z-\eta \right|<r_1=r_1(\psi ,\tau ,\nu ,\eta ,c,\delta )$ where $r_1(\psi ,\tau ,\nu ,\eta ,c,\delta ),$ which has the highest value

$${\displaystyle \frac{(1+\delta )\left(1-2\psi \right)\left(1-\nu \right)c}{\left[1+\nu +2\tau \right]{\mathsf{\Omega }}_s^d\left(1\right)}}{r}^2+{\displaystyle \frac{\left(l\right(l+2-\delta \left)\right(1-2\psi \left)\right(1-\nu \left)\right(1-c)}{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l_o\right)}}{r}^{l+1}\le (1-\delta ),$$

(49)

for $l\ge 2.$ The result is sharp for the function

$${\zeta }_l\left(z\right)={\displaystyle \frac{1}{z-\eta }}+{\displaystyle \frac{(1-2\psi )(1-\nu )c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}(z-\eta )+{\displaystyle \frac{(1-2\psi )(1-\nu )(1-c)}{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)}}{\left(z-\eta \right)}^l,$$

(50)

for some l.

Proof. 

It is sufficient to show that

$$\left|{\displaystyle \frac{(z-\eta ){\zeta }^{{}^{\prime \prime }}\left(z\right)}{{\zeta }^{{}^{\prime }}\left(z\right)}}+2\right|\le 1-\delta (0\le \delta <1) \mathrm{for} 0<\left|z-\eta \right|<r_1(\psi ,\tau ,\nu ,\eta ,c,\delta ).$$

Note that

$$\left|{\displaystyle \frac{(z-\eta ){\zeta }^{{}^{\prime \prime }}\left(z\right)}{{\zeta }^{{}^{\prime }}\left(z\right)}}+2\right|=\left|{\displaystyle \frac{{\displaystyle \frac{2\left(1-2\psi \right)\left(1-\nu \right)c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}{(z-\eta )}^2+{\sum }_{l=2}^{\infty }l\left(l+1\right)a_l{(z-\eta )}^{l+1}}{-1+{\displaystyle \frac{\left(1-2\psi \right)\left(1-\nu \right)c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}{(z-\eta )}^2+{\sum }_{l=2}^{\infty }l{a}_l{(z-\eta )}^{l+1}}}\right|\le 1-\delta$$

$$={\displaystyle \frac{{\displaystyle \frac{2\left(1-2\psi \right)\left(1-\nu \right)c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}{|z-\eta |}^2+{\sum }_{l=2}^{\infty }l\left(l+1\right)a_l{|z-\eta |}^{l+1}}{1-\left({\displaystyle \frac{\left(1-2\psi \right)\left(1-\nu \right)c}{[\left(1+\tau \right)+\left(\nu +\tau \right)]{\mathsf{\Omega }}_s^d\left(1\right)}}{|z-\eta |}^2+{\sum }_{l=2}^{\infty }l{a}_l{|z-\eta |}^{l+1}\right)}}\le 1-\delta$$

$${\displaystyle \frac{{\displaystyle \frac{2\left(1-2\psi \right)\left(1-\nu \right)c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}{r}^2+{\sum }_{l=2}^{\infty }l\left(l+1\right)a_l{r}^{l+1}}{1-{\displaystyle \frac{\left(1-2\psi \right)\left(1-\nu \right)c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}{r}^2-{\sum }_{l=2}^{\infty }l{a}_l{r}^{l+1}}}\le 1-\delta ,$$

(51)

for $0<\left|z-\eta \right|<r$ if, and only if,

$${\displaystyle \frac{2\left(1-2\psi \right)\left(1-\nu \right)c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}{r}^2+\sum _{l=2}^{\infty }l\left(l+1\right)a_l{r}^{l+1}\le (1-\delta )\left(1-{\displaystyle \frac{\left(1-2\psi \right)\left(1-\nu \right)c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}{r}^2-\sum _{l=2}^{\infty }l{a}_l{r}^{l+1}\right)$$

$${\displaystyle \frac{(3-\delta )\left(1-2\psi \right)\left(1-\nu \right)c}{[1+\nu +2\tau ]{\mathsf{\Omega }}_s^d\left(1\right)}}{r}^2+\sum _{l=2}^{\infty }l\left(l+2-\delta \right)a_l{r}^{l+1}\le (1-\delta ).$$

(52)

Since $\zeta \left(z\right)$ is in the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right),$ from (32), we may take

$$a_l={\displaystyle \frac{(1-2\psi )(1-\nu )(1-c){\lambda }_l}{[l\left(1+\tau \right)+\left(\nu +\tau \right)](1+l\psi -\psi ){\mathsf{\Omega }}_s^d\left(l\right)}}(l\ge 2),$$

(53)

where ${\lambda }_l\ge 0(l\ge 2)$ and

$$\sum _{l=1}^{\infty }{\lambda }_l\le 1.$$

(54)

We select the positive integer $l_0=l_0\left(r\right)$ for each fixed r, where ${\displaystyle \frac{l\left(l+2-\delta \right)r^{l+1}}{[l\left(1+\tau \right)+\left(\nu +\tau \right)]\left(1+l\psi -\psi \right){\mathsf{\Omega }}_s^d\left(l\right)}}$ is maximal. Then it follows that

$$\sum _{l=2}^{\infty }l\left(l+2-\delta \right)a_l{r}^{l+1}\le {\displaystyle \frac{(l_o(l_o+2-\delta )(1-2\psi )(1-\nu )(1-c)}{[l_o\left(1+\tau \right)+\left(\nu +\tau \right)](1+l_o\psi -\psi ){\mathsf{\Omega }}_s^d\left(l_o\right)}}{r}^{l_o+1}(l\ge 2).$$

(55)

Then $\zeta \left(z\right)$ is convex of order $\delta$ in $0<\left|z-\eta \right|<r_1(\psi ,\tau ,\nu ,\eta ,c,\delta )$ provided that

$${\displaystyle \frac{(3-\delta )\left(1-2\psi \right)\left(1-\nu \right)c}{[(1+\tau )+\left(\nu +\tau \right)]{\mathsf{\Omega }}_s^d\left(1\right)}}{r}^2+{\displaystyle \frac{(l_o(l_o+2-\delta )(1-2\psi )(1-\nu )(1-c)}{[l_o\left(1+\tau \right)+\left(\nu +\tau \right)](1+l_o\psi -\psi ){\mathsf{\Omega }}_s^d\left(l_o\right)}}{r}^{l_o+1}\le (1-\delta ).$$

(56)

We find the value $r_o=r_o(\psi ,\tau ,\nu ,\eta ,c,\delta )$ and the corresponding integer $l_o\left(r_o\right)$ so that

$${\displaystyle \frac{(3-\delta )\left(1-2\psi \right)\left(1-\nu \right)c}{[(1+\tau )+\left(\nu +\tau \right)]{\mathsf{\Omega }}_s^d\left(1\right)}}{r}_o^2+{\displaystyle \frac{(l_o(l_o+2-\delta )(1-2\psi )(1-\nu )(1-c)}{[l_o\left(1+\tau \right)+\left(\nu +\tau \right)](1+l_o\psi -\psi ){\mathsf{\Omega }}_s^d\left(l_o\right)}}{r}_o^{l_o+1}=(1-\delta ).$$

(57)

Then, this value $r_o$ is the radius of meromorphically convex of order $\delta$ for functions belonging to the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right).$    □

5. Conclusions

The fixed second coefficients of class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right)$ and the q- hypergeometric functions are included in the new class of meromorphic parabolic starlike functions defined in this article. Some features are obtained for the function in the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right)$, including the radius of convexity, closure theorems, and coefficient inequalities.